Prediction of dynamical systems from time-delayed measurements with self-intersections
arxiv(2022)
摘要
In the context of predicting behaviour of chaotic systems, Schroer, Sauer,
Ott and Yorke conjectured in 1998 that if a dynamical system defined by a
smooth diffeomorphism T of a Riemannian manifold X admits an attractor with
a natural measure μ of information dimension smaller than k, then k
time-delayed measurements of a one-dimensional observable h are generically
sufficient for μ-almost sure prediction of future measurements of h. In a
previous paper we established this conjecture in the setup of injective
Lipschitz transformations T of a compact set X in Euclidean space with an
ergodic T-invariant Borel probability measure μ. In this paper we prove
the conjecture for all Lipschitz systems (also non-invertible) on compact sets
with an arbitrary Borel probability measure, and establish an upper bound for
the decay rate of the measure of the set of points where the prediction is
subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and
Yorke related to empirical prediction algorithms as well as algorithms
estimating the dimension and number of required delayed measurements (the
so-called embedding dimension) of an observed system. We also prove general
time-delay prediction theorems for locally Lipschitz or Hölder systems on
Borel sets in Euclidean space.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要